Name: velocity and acceleration in motion graphs
Uploaded: 2026-01-12T12:24:34.000Z
Duration: 1 min 18 s
Channel: JoVE

Imagina un coche que circula por una carretera recta. Su posición a lo largo del tiempo se modela mediante una función cuyo grafo es una curva suave.Esta función rastrea la desplazamiento del coche en relación con un punto de origen fijo a lo largo del tiempo.La tasa instantánea de cambio de posición del coche se mide por su velocidad, mostrando su velocidad y dirección. Se encuentra tomando la primera derivada de la función de posición.El coche se detiene momentáneamente cuando su velocidad es igual a cero. Para la función de posición dada, esta condición de velocidad cero se cumple en momentos específicos.Una velocidad positiva indica que el coche avanza, mientras que una velocidad negativa significa que se mueve hacia atrás hacia su posición inicial.De manera similar, la aceleración mide cómo cambia la velocidad. Se encuentra tomando la derivada de la función de velocidad.El coche acelera cuando la velocidad y la aceleración tienen el mismo signo, y reduce la velocidad cuando tienen signos opuestosEsto muestra cómo la tasa de cambio predice el estado exacto de movimiento del coche en un momento dado más allá de la simple observación.

aplicación de las tasas de cambio

El movimiento de un autom&#xF3;vil a lo largo de una autopista puede examinarse mediante principios clave de c&#xE1;lculo y cinem&#xE1;tica. A medida que el autom&#xF3;vil se desplaza, su posici&#xF3;n var&#xED;a con el tiempo y puede representarse matem&#xE1;ticamente como una funci&#xF3;n del tiempo. Analizar la tasa de cambio de estas cantidades permite medir la velocidad y la aceleraci&#xF3;n, aspectos fundamentales del an&#xE1;lisis del movimiento.
La velocidad describe c&#xF3;mo var&#xED;a la posici&#xF3;n con respecto al tiempo. La velocidad promedio durante un intervalo de tiempo espec&#xED;fico se calcula dividiendo el cambio de posici&#xF3;n entre el cambio correspondiente en el tiempo. Esta medida proporciona una idea general de cu&#xE1;n r&#xE1;pido se desplaza el autom&#xF3;vil, pero no considera las fluctuaciones en el intervalo. Es necesario reducir progresivamente el intervalo de tiempo para medir la velocidad instant&#xE1;nea en un instante dado. Esta velocidad instant&#xE1;nea corresponde a la derivada de la funci&#xF3;n de posici&#xF3;n, que representa la pendiente de la recta tangente en un gr&#xE1;fico posici&#xF3;n-tiempo en un punto dado.
La aceleraci&#xF3;n mide c&#xF3;mo var&#xED;a la velocidad con respecto al tiempo. Al igual que la velocidad, puede evaluarse como un promedio a lo largo de un per&#xED;odo o como un valor instant&#xE1;neo. La aceleraci&#xF3;n media se obtiene dividiendo el cambio de velocidad entre el tiempo transcurrido. Cuando el intervalo de tiempo se aproxima a cero, la aceleraci&#xF3;n en un momento espec&#xED;fico se obtiene derivando la velocidad respecto del tiempo. Esta segunda derivada de la posici&#xF3;n con respecto al tiempo revela la tasa de cambio de la velocidad del autom&#xF3;vil.
Las gr&#xE1;ficas de posici&#xF3;n y velocidad con respecto al tiempo ofrecen informaci&#xF3;n valiosa sobre el movimiento. En un gr&#xE1;fico posici&#xF3;n-tiempo, la pendiente de la recta secante representa la velocidad media, mientras que la pendiente de la recta tangente en un punto determinado muestra la velocidad instant&#xE1;nea. La pendiente de la recta tangente en una gr&#xE1;fica de velocidad-tiempo muestra la aceleraci&#xF3;n, lo que demuestra c&#xF3;mo var&#xED;a la velocidad con el tiempo. Estos m&#xE9;todos gr&#xE1;ficos desempe&#xF1;an un papel crucial en la f&#xED;sica y la ingenier&#xED;a, ya que permiten la interpretaci&#xF3;n y predicci&#xF3;n de escenarios de movimiento reales.

Aplicación de las tasas de cambio

El movimiento de un autom&#xF3;vil a lo largo de una autopista puede examinarse mediante principios clave de c&#xE1;lculo y cinem&#xE1;tica. A medida que ...

Education

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JoVE Core

Calculus

Cálculo

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Unidad

Differentiation Rules

Reglas de derivación

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

Application of Rates of Change

velocity and acceleration in motion graphs

Velocity and acceleration describe how a car moves along a highway. The car’s position changes over time, so it can be written as a function of time. Rates of change help measure motion in a clear mathematical way.

Velocity tells how position changes with time. Average velocity is found by dividing the change in position by the change in time. It gives the overall speed of the car during a time interval, but it does not show changes inside that interval.

To find the velocity at one exact moment, the time interval must get smaller and smaller. This leads to instantaneous velocity. It is the derivative of the position function and matches the slope of the tangent line on a position-time graph.

Acceleration describes how velocity changes over time. Like velocity, it can be measured as an average value over a time period or as an instantaneous value at a specific moment. Average acceleration is the change in velocity divided by the time taken.

When the time interval approaches zero, acceleration at one moment is found by differentiating velocity with respect to time. This is the second derivative of position with respect to time. It shows the rate at which the car’s velocity changes.

Graphs of position and velocity over time help interpret motion. On a position-time graph, the slope of a secant line shows average velocity, and the slope of a tangent line shows instantaneous velocity. On a velocity-time graph, the slope of the tangent line shows acceleration. These graph tools are useful in physics and engineering for understanding and predicting real-world motion.

Velocity and Acceleration in Motion Graphs

velocity-and-acceleration-in-motion-graphs

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application of rates of change

Velocity and acceleration describe how a car moves along a highway. The car’s position changes over time, so it can be written as a function of time. Rates of change help measure motion in a clear mathematical way. Velocity tells how position changes with time. Average velocity is found by dividing the change in position by the change in time. It gives the overall speed of the car during a time interval, but it does not show changes inside that interval. To find the velocity at one exact...

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key terms and concepts

Learn how position-time and velocity-time graphs show average and instantaneous velocity, plus acceleration, using derivatives and slopes in car motion.

Velocity and Acceleration in Motion Graphs | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Velocity and Acceleration in Motion Graphs

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